On implicit extensions in many-valued logic

We consider Kuznetsov's implicit expressibility and its generalizations, when the implicit expressibility language is augmented with the additional disjunction, implication, and negation logical connectives. It is shown that, for each $k\geqslant 3$, the implicit extensions in $P_k$ have the cardinality of the continuum. For each $k\geqslant 3$, we also prove that each of the sets of positively implicit, implicatively implicit, and negatively implicit extensions in $P_k$ contains, respectively, as a proper subset, the set of positively implicit, implicatively implicit, and negatively implicit closed classes. We verify that, for $k\geqslant 2$, the functions of the set $H_k^*$ of homogeneous functions preserving the set $E_{k-1}$ can be used for producing implicatively implicit and negatively implicit extensions without changing the result.